Where are the singularities of f(z) = Log(2+tan(z)) located?

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SUMMARY

The singularities of the function f(z) = Log(2 + tan(z)) are determined by analyzing the components of the logarithmic function. The expression can be rewritten as Log(2 + tan(z)) = ln(abs(2 + tan(z))) + i*Arg(2 + tan(z)). The logarithm is undefined when its argument equals zero, which occurs when 2 + tan(z) = 0. This leads to singularities at points where tan(z) = -2, specifically at z = arctan(-2) + nπ, where n is any integer. The argument of the logarithm also introduces branch cuts, particularly along the negative imaginary axis, affecting the definition of the function.

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I need to find the locations of the singularity of f(z) = Log(2+tan(z)).

So far I have looked at the function in its alternate form

Log(2+tan(z)) = ln(abs(2+tan(z))) + i*Arg(2+tan(z))

If I remember correctly the first part is simple and cannot equal zero.

Now I think the second part is defined on the cut 0 to 2∏ which eliminates the negative imaginary axis.
 
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first determine what x makes log(x) a singularity then determine how x can be assigned that value by2+tan(z)
 

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